If \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar vectors and \( \lambda \) is a real number, then the vectors
\[
\vec{a} + 2\vec{b} + 3\vec{c}, \quad \lambda\vec{b} + 4\vec{c}, \quad (2\lambda - 1)\vec{c}
\]
are non-coplanar for:
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For non-coplanar vector problems:
\begin{itemize}
\item Use scalar triple product.
\item Convert vectors into coefficient matrix.
\item Nonzero determinant ⇒ non-coplanar.
\end{itemize}